What is the general solution for sin(2x)+sin(x)-sin(3x)=0 ?

The general solution for sin(2x)+sin(x)-sin(3x)=0 is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3

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sin(2x)+sin(x)-sin(3x)=0

Solution

sin (2 x) + sin (x) - sin (3 x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 24 0^{\circ} n, x = 12 0^{\circ} + 24 0^{\circ} n

Solution steps

sin (2 x) + sin (x) - sin (3 x) = 0

Rewrite using trig identities

sin (2 x) - sin (3 x) + sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (x) cos (x) - sin (3 x) + sin (x)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 cos (x) sin (x) + 2 sin (\frac{x - 3 x}{2}) cos (\frac{x + 3 x}{2})

2 sin (\frac{x - 3 x}{2}) cos (\frac{x + 3 x}{2}) = - 2 cos (2 x) sin (x)

2 sin (\frac{x - 3 x}{2}) cos (\frac{x + 3 x}{2})

\frac{x - 3 x}{2} = - x

\frac{x - 3 x}{2}

Add similar elements:

x - 3 x = - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= - x

= 2 sin (- x) cos (\frac{x + 3 x}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= 2 cos (\frac{x + 3 x}{2}) (- sin (x))

Remove parentheses:

(- a) = - a

= - 2 cos (\frac{x + 3 x}{2}) sin (x)

\frac{x + 3 x}{2} = 2 x

\frac{x + 3 x}{2}

Add similar elements:

x + 3 x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= - 2 cos (2 x) sin (x)

= 2 cos (x) sin (x) - 2 cos (2 x) sin (x)

- 2 cos (2 x) sin (x) + 2 cos (x) sin (x) = 0

Factor

- 2 cos (2 x) sin (x) + 2 cos (x) sin (x) : 2 sin (x) (- cos (2 x) + cos (x))

- 2 cos (2 x) sin (x) + 2 cos (x) sin (x)

Rewrite as

= - 2 sin (x) cos (2 x) + 2 sin (x) cos (x)

Factor out common term

2 sin (x)

= 2 sin (x) (- cos (2 x) + cos (x))

2 sin (x) (- cos (2 x) + cos (x)) = 0

Solving each part separately

sin (x) = 0 or - cos (2 x) + cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- cos (2 x) + cos (x) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

- cos (2 x) + cos (x) = 0

Rewrite using trig identities

- cos (2 x) + cos (x)

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2})

Simplify

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2}) : 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2})

Add similar elements:

x + 2 x = 3 x

= - 2 sin (\frac{3 x}{2}) sin (\frac{x - 2 x}{2})

\frac{x - 2 x}{2} = - \frac{x}{2}

\frac{x - 2 x}{2}

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{x}{2}

= - 2 sin (\frac{3 x}{2}) sin (- \frac{x}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= - 2 (- sin (\frac{x}{2})) sin (\frac{3 x}{2})

Apply rule

- (- a) = a

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{3 x}{2}) = 0

General solutions for

sin (\frac{3 x}{2}) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

Solve

\frac{3 x}{2} = 0 + 2 πn : x = \frac{4 πn}{3}

\frac{3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{2} = 2 πn

Multiply both sides by

2

\frac{3 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 2 πn

Simplify

3 x = 4 πn

3 x = 4 πn

Divide both sides by

3

3 x = 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{4 πn}{3}

Simplify

x = \frac{4 πn}{3}

x = \frac{4 πn}{3}

Solve

\frac{3 x}{2} = π + 2 πn : x = \frac{2 π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

3 x = 2 π + 4 πn

3 x = 2 π + 4 πn

Divide both sides by

3

3 x = 2 π + 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3}

Simplify

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) = 0

General solutions for

sin (\frac{x}{2}) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Solve

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

Solve

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}, x = 4 πn, x = 2 π + 4 πn

Merge Overlapping Intervals

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)+sin(x)-sin(3x)=0 ?
The general solution for sin(2x)+sin(x)-sin(3x)=0 is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3